Major sudden stratospheric warmings (SSWs) are the most important phenomena of the wintertime boreal stratospheric variability. During SSWs, the polar temperature increases abruptly, and easterlies prevail in the stratosphere. Their effects extend farther from the polar stratosphere, affecting near-surface circulation. According to observations, SSWs are not equally distributed in time, with decades experiencing very few events, while others experiencing SSWs almost every winter. Some sources of this SSW multidecadal variability can be traced back to sea surface temperature changes. Here, we investigate the effects of Pacific decadal variability (PDV) and Atlantic multidecadal variability (AMV) on SSWs. We use for the first time a large ensemble of historical experiments to examine the modulation of the frequency, tropospheric precursors, and impact of SSWs by the PDV and AMV. We find a strong impact of the PDV on the occurrence of SSWs, with a higher SSW frequency for the positive phase of the PDV. This PDV influence is mediated by constructive interference of PDV anomalies with tropospheric stationary waves. The main effect of AMV is, instead, a modulation of the tropospheric response to SSWs, a finding that can be useful for predicting the tropospheric fingerprint of SSWs.